Chapter 9

The hypotenuse of a right triangle is 3 feet. If one leg is 1 foot, find the degree measure of each angle.

Solve each triangle. $$ B=20^{\circ}, \quad C=70^{\circ}, \quad a=1 $$

A right triangle has a hypotenuse of length 8 inches. If one angle is \(35^{\circ},\) find the length of each leg.

A right triangle has a hypotenuse of length 10 centimeters. If one angle is \(40^{\circ},\) find the length of each leg.

If two angles and the included side are given, the third angle is easy to find. Use the Law of sines to show that the area \(K\) of a triangle with side \(a\) and angles \(A, B,\) and \(C\) is $$K=\frac{a^{2} \sin B \sin C}{2 \sin A}$$

Solve each triangle. $$ a=6, \quad b=11, \quad c=12 $$

At \(10 \mathrm{AM}\) on April 26,2018 , a building 300 feet high cast a shadow 50 feet long. What was the angle of elevation of the Sun?

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=9, \quad c=4, \quad B=115^{\circ} $$

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=2, \quad c=1, \quad A=120^{\circ} $$

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=7, \quad b=14, \quad A=30^{\circ} $$

A security camera in a neighborhood bank is mounted on a wall 9 feet above the floor. What angle of depression should be used if the camera is to be directed to a spot 6 feet above the floor and 12 feet from the wall?

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=2, \quad c=3, \quad B=40^{\circ} $$

In Problems 33-44, solve each triangle. $$ B=20^{\circ}, C=75^{\circ}, b=5 $$

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=4, \quad c=6, \quad B=20^{\circ} $$

Graph each function by adding y-coordinates. $$ g(x)=\cos (2 x)+\cos x $$

Solve each triangle. $$ A=50^{\circ}, B=55^{\circ}, c=9 $$

Once the box begins to slide and accelerate, kinetic friction acts to slow the box with a coefficient of kinetic friction \(\mu_{k}=0.1 .\) The raised end of the surface can be lowered to a point where the box continues sliding but does not accelerate. The critical angle at which this happens, \(\theta_{c}^{\prime},\) can be found from the equation \(\tan \theta_{c}^{\prime}=\mu_{k}\) (a) What is this critical angle for the box? (b) If the box is \(5 \mathrm{ft}\) from the pivot point, at what height will the box stop accelerating?

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=3, \quad b=7, \quad A=70^{\circ} $$

Mixed Practice In Problems \(35-40,\) (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ f(x)=\sin (2 x) \sin x $$

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=8, \quad c=3, \quad C=125^{\circ} $$

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=4, \quad c=5, \quad B=95^{\circ} $$

(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ G(x)=\cos (4 x) \cos (2 x) $$

A DC-9 aircraft leaves Midway Airport from runway 4 RIGHT, whose bearing is \(\mathrm{N} 40^{\circ} \mathrm{E} .\) After flying for \(\frac{1}{2}\) mile, the pilot requests permission to turn \(90^{\circ}\) and head toward the southeast. The permission is granted. After the airplane goes 1 mile in this direction, what bearing should the control tower use to locate the aircraft?

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=7, \quad c=3, \quad C=12^{\circ} $$

(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ h(x)=\cos (2 x) \cos (x) $$

Find the area of the segment of a circle whose radius is 5 inches, formed by a central angle of \(40^{\circ}\).

A ship leaves the port of Miami with a bearing of \(\mathrm{S} 80^{\circ} \mathrm{E}\) and a speed of \(15 \mathrm{knots}\). After 1 hour, the ship turns \(90^{\circ}\) toward the south. After 2 hours, maintaining the same speed, what is the bearing to the ship from the port?

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=4, \quad c=5, \quad B=40^{\circ} $$

(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ H(x)=2 \sin (3 x) \cos (x) $$

The dimensions of a triangular lot are 100 feet by 50 feet by 75 feet. If the price of the land is \(\$ 3\) per square foot, how much does the lot cost?

In Problems \(41-46,\) an object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=25, \quad a=10, \quad b=0.7, \quad T=5 $$

The angle of inclination from the base of the John Hancock Center to the top of the main structure of the Willis Tower is approximately \(10.3^{\circ}\). If the main structure of the Willis Tower is 1451 feet tall, how far apart are the two skyscrapers? Assume the bases of the two buildings are at the same elevation.

An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=20, \quad a=15, \quad b=0.75, \quad T=6 $$

An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=30, \quad a=18, \quad b=0.6, \quad T=4 $$

An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=15, \quad a=16, \quad b=0.65, \quad T=5 $$

Find the area of the shaded region enclosed in a semicircle of diameter 10 inches. The length of the chord \(P Q\) is 8 inches. [Hint: Triangle \(P Q R\) is a right triangle.]

An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=10, \quad a=5, \quad b=0.8, \quad T=3 $$

Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is \(30^{\circ},\) and from a second position 40 feet farther along this path it is \(20^{\circ} .\) What is the height of the tree?

A loading ramp 10 feet long that makes an angle of \(18^{\circ}\) with the horizontal is to be replaced by one that makes an angle of \(12^{\circ}\) with the horizontal. How long is the new ramp?

In Problems \(47-52,\) the function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-20 e^{-0.7 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} t\right) $$

Completed in 1902 in New York City, the Flatiron Building is triangular shaped and bounded by 22nd Street, Broadway, and 5th Avenue. The building measure approximately 87 feet on the 22 nd Street side, 190 feet on the Broadway side, and 173 feet on the 5 th Avenue side. Approximate the ground area covered by the building.

Adam must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles. A second flight option flies first to Kansas City and then connects to St. Louis. The bearing from Oklahoma City to Kansas City is N29.6 \({ }^{\circ} \mathrm{E}\), and the bearing from Oklahoma City to St. Louis is N57.7 \(^{\circ}\) E. The bearing from St. Louis to Oklahoma City is \(S 57.7^{\circ}\) W, and the bearing from St. Louis to Kansas City is \(\mathrm{N} 79.4^{\circ} \mathrm{W}\). How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct flight?

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-20 e^{-0.8 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.64}{1600}} t\right) $$

The Bermuda Triangle is roughly defined by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. The distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton are approximately \(1028,1046,\) and 965 miles, respectively. Ignoring the curvature of Earth, approximate the area of the Bermuda Triangle.

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-30 e^{-0.6 t / 80} \cos \left(\sqrt{\left(\frac{2 \pi}{7}\right)^{2}-\frac{0.36}{6400}} t\right) $$

There is a Heron-type formula that can be used to find the area of a general quadrilateral. $$K=\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^{2} \theta}$$ where \(a, b, c,\) and \(d\) are the side lengths, \(\theta\) is half the sum of two opposite angles, and \(s\) is half the perimeter. Show that if a triangle is considered a quadrilateral with one side equal to \(0,\) Bretschneider's Formula reduces to Heron's Formula.

A major league baseball diamond is actually a square 90 feet on a side. The pitching rubber is located 60.5 feet from home plate on a line joining home plate and second base. (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates the bearing of the ship is \(\mathrm{N} 55^{\circ} \mathrm{E} ;\) the call to Station Baker indicates the bearing of the ship is \(\mathrm{S} 60^{\circ} \mathrm{E}\). (a) How far is each station from the ship? (b) If a helicopter capable of flying 200 miles per hour is dispatched from the station nearest the ship, how long will it take to reach the ship?

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-30 e^{-0.5 t / 70} \cos \left(\sqrt{\left(\frac{\pi}{2}\right)^{2}-\frac{0.25}{4900}} t\right) $$

(a) Show that the area of a regular dodecagon (12-sided polygon) is given by \(K=3 a^{2} \cot \frac{\pi}{12}\) or \(K=12 r^{2} \tan \frac{\pi}{12}\) where \(a\) is the length of one of the sides and \(r\) is the radius of the inscribed circle.